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The existence of the density of the Euler characteristic was proved by Gusein-Zade, see [1]. If a quasiperiodic set is discrete, then the density of its Euler characteristic equals the density of the set itself, which exists by Soprunovas theorem [2].

Theorem. The densities of the Betti numbers of a quasiperiodic subanalytic set exist.

The paper [3] contains a refined version of this theorem, which, in particular, explains, how to compute the answer approximately. However, note that the densities of the Betti numbers of L are not shown to depend analytically on M, which is the case for the Euler characteristic. In addition to ideas from [1] and [2], the proof is based on the following simple fact, applied to a certain infinite cell decomposition of the quasiperiodic set.

If 1,..., a are the cells of a cell complex K, then there are two options for every cell i of dimension k: attaching i to the union i-1j j=either increases the k-th Betti number of the union by 1, or decreases its (k - 1)-th Betti number by 1. Thus, the l-th Betti number of K equals the number of l-dimensional cells of the first kind minus the number of (l + 1)-dimensional cells of the second kind.

References [1] Gusein-Zade S.M. On the topology of quasiperiodic functions. // Amer. Math. Soc. Transl. Ser. 2. 1999. 197. Pseudoperiodic topology. p. 1 7.

[2] Soprunova E. Zeros of Systems of Exponential Sums and Trigonometric Polynomials. // Moscow Math. J. 2006. 6:1. p. 153 168.

[3] Esterov A. Densities of topological invariants of subanalytic quasiperiodic sets. // To appear in Izvestiya RAN.

Confluence of eigenvalues and resonant Stokes operators Glutsyuk Alexey A. (France) cole Normale Suprieure de Lyon aglutsyu@umpa.ens-lyon.fr The talk deals with linear ordinary differential equations in complex time with irregular singularity at 0:

A(t) = z, z Cn, A(0) = 0.

tk+The analytic classification invariants of these equations are formal normal form and Stokes operators, which are transition operators between appropriate solution bases (called canonical sectorial solution bases). The irregular singularity is called resonant, if the higher term matrix A(0) in the right-hand side has multiple eigenvalue.

We study a resonant irregular singularity as a limit of degenerating nonresonant singularities. The unfolding under consideration of a resonant singularity is a family of linear equations depending on a real parameter. The perturbed equation is nonresonant and some (distinct) eigenvalues of its higher term matrix are confluent to a multiple eigenvalue. The (nonperturbed) resonant equation and its unfolding should satisfy some genericity assumptions.

The main result says that appropriate canonical sectorial solution bases of the perturbed (nonresonant) equation tend to some canonical sectorial solution bases of the nonperturbed (resonant) equation. This implies that appropriate Stokes operators of the perturbed equation tend to some Stokes operators of the nonperturbed equation.

The results of the talk extend previous analogous results of the speaker, J.-P. Ramis, A. Duval, C. Zhang, R. Schfke (see [1] and the references therein) that study an irregular singularity as a limit of confluenting Fuchsian singularities (following an idea due to V. I. Arnold and J.-P. Ramis (1980-ths) to study Stokes operators as limit monodromy data of Fuchsian equation).

References [1] Glutsyuk A. A., On the monodromy group of confluent linear equations, Moscow Math. J., Vol. 5 (2005), No. 1, 6790.

On the structure of 1 : 4 resonances in Henon-like maps Gonchenko M. S. (Spain) Universitat Politcnica de Catalunya marina.gonchenko@upc.edu We observe results of [1] on bifurcations of fixed points with multipliers ei/2 (the so-called 1:4 resonances) for certain Hnon-like maps in two main cases:

1) the generalized Hnon maps (GHM) x = y, y = M1 - M2x - y2 + Rxy + Sy3;

2) the cubic Hnon maps (CHM) x = y, y = M1 - Bx + M2y y3.

Here (x, y) R2, M1 and M2 are parameters, R and S are small coefficients.

First, we deal with conservative GHMs (M2 1 and R 0) and CHMs (B 1). In the case of GHMs the conservative bifurcations are nondegenerate if S = 0 and they are essentially different depending on the sign of S. A two-parameter analysis of the bifurcations at the critical moment S = 0 is also given. In CHMs the structure of the conservative 1 : 4 resonances is nondegenerate always for the cubic map with +, whereas, for the cubic map with - a degenerate situation is observed for M1 = 16/27, M2 = 1/3.

In the case of nonconservative GHMs, we find conditions of nondegeneracy of the corresponding 1 : 4 resonances and give a description of accompanying bifurcations.

This work is supported by the grant of Spanish grant FPU AP20054492 of Programa de becas FPU del Ministerio de Educacin y Ciencia.

References [1] M.S. Gonchenko. On the structure of 1:4 resonances in Henon maps.

// Int.J. Bifurcation and Chaos. 2005. N15(11). p. 36533660.

On fractal dimension of oscillatory motions Gorodetski A. S. (Russia, USA) Moscow Independent University UC Irvine asgor@mccme.ru A motion of the 3 body problem is called oscillatory if limsup of maximal distance among the bodies is infinity as time tends to infinity and liminf is finite. V.M.Alexeev [1] explained the existence of the oscillatory motions in Sitnikov model (one of the restricted versions of the three body problem) using methods of hyperbolic dynamics. Kolmogorov conjectured that the set of oscillatory motions has zero measure.

In our joint work with V.Kaloshin [2] we show that in many cases the set of oscillatory motions in the 3 body problem has maximal Hausdorff dimension. Proof relies on investigation of area-preserving Henon family, persistent homoclinic tangencies, and splitting of separatrices. Namely, consider the Sitnikov problem. It is a special case of the restricted three body problem where the two primaries with equal masses are moving in an elliptic orbits of the two body problem, and the infinitesimal mass is moving on the straight line orthogonal to the plane of motion of the primaries which passes through the center of mass. Eccentricity e0 of orbits of primaries is a parameter. After some change of coordinates (McGehee transformation) the infinity can be considered as a degenerate saddle with smooth invariant manifolds that correspond to parabolic motions (the orbit tends to infinity with zero limit velocity). Stable and unstable manifolds coincide in the case of circular (e0 = 0) Sitnikov problem.

Dankowicz and Holmes [3] showed that for non-zero eccentricity invariant manifolds have a point of transverse intersection. This leads to the existence of homoclinic tangencies and appearance of all the phenomena that can be encountered in the conservative homoclinic bifurcations.

P. Duarte [4] showed that in area preserving case existence of homoclinic tangencies leads to phenomena similar to Newhouse phenomena, where sinks are replaced by elliptic islands. We prove a stronger one parameter version of the Duartes result. Namely, consider a generic unfolding of a quadratic homoclinic tangency associated with a saddle Pof an area preserving map f0. There is an open set U in the space of parameters such that for every U the map f has a hyperbolic set with persistent homoclinic tangencies. Moreover, for every from some residual subset of U the map f has an invariant transitive closed set H such that the set H is accumulated by fs elliptic points, dimHH = 2, dimH{x H| P (x) (x)} = 2, where P is the unique fixed point near P0.

In particular, existence of these sets of large Hausdorff dimension implies that there is an open set U, 0 U, in the space of parameters of the Sitnikov problem such that for parameters from some residual subset of U the set of oscillatory orbits has full Hausdorff dimension. Similar statement holds for the planar circular restricted three body problem. The existence of transversal homoclinic points in the latter case was established in [5], [6].

References [1] Alexeyev V., Sur lallure finale du mouvement dans le probleme des trois corps. // Actes du Congres International des Mathematiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, Tome 2, p. 893 907.

[2] Gorodetski A., Kaloshin V. Hausdorff dimension of oscillatory motions in the restricted planar circular three body problem and in Sitnikov problem, in preparation.

[3] Dankowicz H., Holmes P. The existence of transverse homoclinic points in the Sitnikov problem. // J. Differential Equations. 1995.

vol. 116, no. 2, p.468 483.

[4] Duarte P. Abundance of elliptic isles at conservative bifurcations.

// Dynamics and Stability of Systems. 1999, vol. 14, no. 4, p.

339 356.

[5] Llibre J., Simo C., Oscillatory solutions in the planar restricted three-body problem. // Math. Ann. 1980, vol. 248, p. 153 184.

[6] Xia J., Melnikov method and transversal homoclinic points in the restricted three-body problem. // J. of Diff. Equations. 1992, vol.

96, no. 1, p. 170 184.

On the local Picard group Hamm Helmut A. (Germany) University of Muenster hamm@math.uni-muenster.de Let X be a complex analytic space. The Picard group P ic X of X is the group of isomorphism classes of holomorphic line bundles on X. Recall that P ic X H1(X, OX) where OX is the sheaf of nowhere vanishing holomorphic functions.

In particular let X be a suitable representative of a germ (X, x) of a complex space embedded in Cn. We want to prove a Lefschetz theorem for the local Picard group, i.e. compare P ic(X \{x}) and P ic(X H \{x}) where H is a hyperplane through x which defines a divisor on X.

The ingredients in the hypothesis are the depth of OX and the rectified cohomological depth rcd X with respect to the constant sheaf ZX, see also [4], because of the exponential sequence 0 - ZX - OX - OX - Theorem 1: If depth OX 3 (resp. 4) and rcd(X \ {x}) 4 (resp.

5) the natural mapping P ic(X \ {x}) - P ic(X H \ {x}) is injective (resp. bijective).

The depth condition on the whole space X is quite strong. It can be weakened if we replace X H \ {x} by a suitable neighbourhood U of this space in X \ {x}:

Theorem 2A: If depth OX\{x} 3 (resp. 4) and rcd(X \ {x}) (resp. 5) the mapping P ic(X \ {x}) - P ic U is injective (resp. bijective).

In algebraic geometry it is usual to regard the formal completion X of X along X H as a substitute for a tubular neighbourhood of X H in X. We have a corresponding analogue of Theorem 2A:

Theorem 2B: If depth OX\{x} 3 (resp. 4) and rcd(X \ {x}) (resp. 5) the mapping P ic(X \ {x}) - P ic(X \ {x}) is injective (resp.

bijective).

Furthermore it is possible to compare line bundles (and even vector bundles) on neighbourhoods of X H \ {x} in X \ {x} which are open in the usual resp. in the Zariski topology.

The techniques of proof are of a quite diverse nature. From the topological side we need a local Lefschetz theorem as in [1]. The depth condtions for the structural sheaf are used in order to have vanishing resp. coherence properties for the local cohomology, cf. [2]. For Theorem 2 we need an extension theorem for coherent analytic sheaves on ring domains, cf. [3].

Global Lefschetz theorems for the Picard group, i.e. on projective varieties, have been proved in [4].

In the framework of algebraic geometry, A.Grothendieck has studied the Picard group in the local and global case [5]. His work has stimulated the present one, partially it inspired the methods, too.

References [1] Hamm H.A., L D.T.: Rectified homotopical depth and Grothendieck conjectures. In: The Grothendieck Festschrift vol. II, pp.

311-351. Birkhuser, Boston 1990.

[2] Bnic C., Stnil O.: Algebraic methods in the Global Theory of Complex Spaces. John Wiley, London 1976.

[3] Siu Y.-T.: Techniques of extension of analytic objects. Marcel Dekker, N.Y. 1974.

[4] Hamm H.A., L D.T.: A Lefschetz theorem on the Picard group of complex projective varieties. In: Singularities in Geometry and Topology, pp. 640-660. World Sc. Publ., Singapore 2007.

[5] Grothendieck A.: Cohomologie locale des faisceaux cohrents et thormes de Lefschetz locaux et globaux (SGA II). Masson & Cie., Paris/ North Holland, Amsterdam 1968.

Formal solutions to limit cycles of polinomial differential equations. An approach to solution of Hilberts 16th Problem Hernandez Rosales Manuel (Mexico) Paralaje, Mexico City mhernandez@paralaje.net In this talk I will show you a general method for to obtain formal solutions to periodical solutions of polynomial differential equations. The solutions, given in a series form, are in terms of coefficients that are solution of an set of infinite algebraic equations. These algebraic equations determine all the formal periodic solutions of these polynomial differential equations. The existence of possible limit cycles are represented by isolated solutions of the algebraic equations.

This method then gives an upper bound of Hilbert numbers for the Hilberts 16th Problem for any n when we know the number of isolated solutions of the algebraic equations.

If an set of coefficients are such that the infinite dimensional vector formed by them live in a certain Hilbert space and the coeffcients are an isolated solution of the infinite algebraic equations then the formal solution become a real limit cycle. This gives the possibilitie of total solution of the Hilberts 16th problem.

References [1] V.I. Arnold, Mathematical Methods of Classical Mechanics. // Springer 1989 p. 399[2] Yu. Ilyashenko, Centennial History of Hilberts 16th Problem // Bull. Amer. Math. Soc. 39 (2002), 301-354.

Non-attracting attractors Ilyashenko Yu. S. (Russia) Moscow State University Independent University of Moscow Cornell University, US Steklov Mathematical Institute One of the major problems in the theory of dynamical systems is the study of the limit behavior of solutions. It is a general belief that after a long time delay the observer will see the orbits that belong to this or that type of attractor. Therefore, knowledge of the attractor of the system predicts the long time behavior of solutions.

In the present talk we develop an opposite point of view. Namely, we describe dynamical systems whose attractors have a large part which is in a sense unobservable. This motivated a notion of -attractor. It is a set (not necessary uniquely defined) near which almost all the orbits spend in average more than 1 - part of the future time. We discuss the effect of drastic non-coincidence of actual attractor and -attractor. For sufficiently small, like 10-30, the difference between actual attractors and attractors is unobservable in the computer and physical experiments.

Therefore, -attractors with small have a chance to replace actual attractors in applications.

This is a joint work with Andrei Negut, junior student of Princeton University.

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